Terminology

Here we look at the terminology such as geometries, spaces, models, projections and transforms. Its quite difficult when we start dealing with non-Euclidean geometries because we use similar terminology that we are used to in conventional Euclidean space but the terms can have slightly different properties. For example, the concept of a 'line' can look different in different geometries:

Geometries and Spaces

Rienmannian geometry defines spaces generally in terms of manifolds, here we are interested in homogeneous, isotropic spaces which have no preferred points or directions, examples are:

Minkowski Metric

If we change the definition of 'distance' slightly we create a space with different properties.

In Euclidian space the distance from the origin is given by √(x²+y²+z²) but in Minkowski space we change the definition of distance to √(x²+y²+z²-t²). This gives lots of interesting properties which we look at on this page.

Lie Group

Groups are usually defined over a set but in the case of Lie groups they are defined over manifolds. That is like taking a group, unplugging it from the set structure and instead plugging in the manifold.

So a Lie group is a combination
of infinite groups and calculus. For example 'infinitesimal
elements' allow us to build rotations by integrating some
infinitesimal rotation. A group that has infinitesimal generators is called a continuous group.

It is useful to be able to linearise the group, for instance taking the exponent of the group, this linearised version is known as its Lie algebra , this takes a group with one operation and creates an algebra with both a addition and a multiplication operation.

Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.

Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. This book stresses the Geometry in Geometric Algebra, although it is still very mathematically orientated. Programmers using this book will need to have a lot of mathematical knowledge. Its good to have a Geometric Algebra book aimed at computer scientists rather than physicists. There is more information about this book here.